IRREDUCIBLE (MATHEMATICS)

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In mathematics, the term '''irreducible''' is used in several ways.

★ In abstract algebra, 'irreducible' can be an abbreviation for irreducible element; for example an irreducible polynomial.

★ Absolutely irreducible is a term applied to mean irreducible, even after any finite extension of the field of coefficients. It applies in various situations, for example to irreducibility of a linear representation, or of an algebraic variety; where it means just the same as ''irreducible over an algebraic closure''.

★ In commutative algebra, a commutative ring ''R'' is 'irreducible' if its prime spectrum, that is, the topological space Spec ''R'', is an irreducible topological space.

★ A directed graph is 'irreducible' if, given any two vertices, there exists a path from the first vertex to the second. A digraph is irreducible if and only if its adjacency matrix is irreducible.

★ In a related notion, a matrix is 'irreducible' if it is not similar to a block upper triangular matrix via a permutation. (Replacing non-zero entries in the matrix by one, and viewing the matrix as an adjacency matrix of a graph, the matrix is irreducible if and only if the graph is.)

★ Also, a Markov chain is 'irreducible' if there is a non-zero probability of transitioning from any state to any other state.

★ In the theory of manifolds, an ''n''-manifold is 'irreducible' if any embedded (''n''−1)-sphere bounds an embedded ''n''-ball. Implicit in this definition is the use of a suitable category, such as the category of differentiable manifolds or the category of piecewise-linear manifolds.

The notions of irreducibility in algebra and manifold theory are related. An ''n''-manifold is called prime, if it cannot be written as a connected sum of two ''n''-manifolds (neither of which is an ''n''-sphere). An irreducible manifold is thus prime, although the converse does not hold. From an algebraist's perspective, prime manifolds should be called "irreducible"; however, the topologist (in particular the 3-manifold topologist) finds the definition above more useful. The only compact, connected 3-manifolds that are prime but not irreducible are the trivial 2-sphere bundle over ''S''¹ and the twisted 2-sphere bundle over ''S''¹. See, for example, Prime decomposition (3-manifold).

★ In representation theory, an 'irreducible representation' is a nontrivial representation with no nontrivial subrepresentations. Similarly, an 'irreducible module' is another name for a simple module.

★ A topological space is 'irreducible' if it is not the union of two proper closed subsets. This notion is used in algebraic geometry, where spaces are equipped with the Zariski topology; it is not of much significance for Hausdorff spaces. See also irreducible component, algebraic variety.

★ In universal algebra, 'irreducible' can refer to the inability to represent an algebraic structure as a composition of simpler structures using a product construction; for example subdirectly irreducible.

★ A 3-manifold is P²-irreducible if it is irreducible and contains no 2-sided $mathbb\; RP^2$ (real projective plane).

★ An Irreducible fraction (or 'fraction in lowest terms') is a vulgar fraction in which the numerator and denominator are smaller than those in any other equivalent fraction.